ABSTRACT
FIGURE 8.1: Instantaneous correlation between Libors in a 2-factor LMM. k1 = 0.25, k2 = 0.04, θ1 = 100%, θ2 = 50% and ρ = −20%.
B(z, t, T ) = 1− e−z(T−t)
z
A(t, T ) is left unspecified as we don’t need to know its expression. As
Li(t) = 1 τi
( PtTi−1 PtTi
− 1 )
(8.3)
we deduce that the Libor Li(t) has the following dynamics in the forward measure Pi
dLi(t) = σa(t)(τiLi(t) + 1) ((−e−a(Ti−t) + e−a(Ti−1−t)
aτi
) dWa+
σb(t) σa(t)
(−e−b(Ti−t) + e−b(Ti−1−t) bτi
) dWb
) This SDE is easily obtained by applying Itoˆ’s lemma on (8.3). We don’t need to care about the drift term as it must cancel at this end in the forward measure Pi. Let us assume that the ratio σb(t)σa(t) ≡ θ̂b is a constant. If we choose new parameters ({kε, θε}ε=a,b) such as (θ̂a ≡ 1)
(−e−εTi + e−εTi−1 ετi
) ;
equivalent to
kε = ε
θε = θ̂ε
( 1− e−ετi
ετi
) ≈ θ̂ε
HW2 can be rewritten as (σ(t) ≡ σa(t))
dLi(t) = σ(t)(τiLi(t) + 1) ( θae −ka(Ti−1−t)dWa + θbe−kb(Ti−1−t)dWb
) The correlation structure is identical to (8.1) with r = 2. Once the parameters kp, θp, ρ¯pq have been chosen, we can calibrate the Libor volatilities σi(t) to at-the-money swaptions.
